Optimal. Leaf size=404 \[ -\frac{8 a^{13/4} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (13 \sqrt{a} B-77 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{3003 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^2 \sqrt{e x} \sqrt{a+c x^2} (13 a B-77 A c x)}{3003 c}-\frac{16 a^{13/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{39 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^3 A e x \sqrt{a+c x^2}}{39 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 \sqrt{e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{2145 c}-\frac{4 a \sqrt{e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{9009 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c} \]
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Rubi [A] time = 0.486871, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {833, 815, 842, 840, 1198, 220, 1196} \[ -\frac{8 a^{13/4} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (13 \sqrt{a} B-77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3003 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^2 \sqrt{e x} \sqrt{a+c x^2} (13 a B-77 A c x)}{3003 c}-\frac{16 a^{13/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{39 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^3 A e x \sqrt{a+c x^2}}{39 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 \sqrt{e x} \left (a+c x^2\right )^{5/2} (13 a B-165 A c x)}{2145 c}-\frac{4 a \sqrt{e x} \left (a+c x^2\right )^{3/2} (39 a B-385 A c x)}{9009 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 833
Rule 815
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \sqrt{e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{2 \int \frac{\left (-\frac{1}{2} a B e+\frac{15}{2} A c e x\right ) \left (a+c x^2\right )^{5/2}}{\sqrt{e x}} \, dx}{15 c}\\ &=-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{8 \int \frac{\left (-\frac{13}{4} a^2 B c e^3+\frac{165}{4} a A c^2 e^3 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{429 c^2 e^2}\\ &=-\frac{4 a \sqrt{e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{32 \int \frac{\left (-\frac{117}{8} a^3 B c^2 e^5+\frac{1155}{8} a^2 A c^3 e^5 x\right ) \sqrt{a+c x^2}}{\sqrt{e x}} \, dx}{9009 c^3 e^4}\\ &=-\frac{8 a^2 \sqrt{e x} (13 a B-77 A c x) \sqrt{a+c x^2}}{3003 c}-\frac{4 a \sqrt{e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{128 \int \frac{-\frac{585}{16} a^4 B c^3 e^7+\frac{3465}{16} a^3 A c^4 e^7 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{135135 c^4 e^6}\\ &=-\frac{8 a^2 \sqrt{e x} (13 a B-77 A c x) \sqrt{a+c x^2}}{3003 c}-\frac{4 a \sqrt{e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{\left (128 \sqrt{x}\right ) \int \frac{-\frac{585}{16} a^4 B c^3 e^7+\frac{3465}{16} a^3 A c^4 e^7 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{135135 c^4 e^6 \sqrt{e x}}\\ &=-\frac{8 a^2 \sqrt{e x} (13 a B-77 A c x) \sqrt{a+c x^2}}{3003 c}-\frac{4 a \sqrt{e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac{\left (256 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{585}{16} a^4 B c^3 e^7+\frac{3465}{16} a^3 A c^4 e^7 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{135135 c^4 e^6 \sqrt{e x}}\\ &=-\frac{8 a^2 \sqrt{e x} (13 a B-77 A c x) \sqrt{a+c x^2}}{3003 c}-\frac{4 a \sqrt{e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac{\left (16 a^{7/2} \left (13 \sqrt{a} B-77 A \sqrt{c}\right ) e \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{3003 c \sqrt{e x}}-\frac{\left (16 a^{7/2} A e \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{39 \sqrt{c} \sqrt{e x}}\\ &=\frac{16 a^3 A e x \sqrt{a+c x^2}}{39 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{8 a^2 \sqrt{e x} (13 a B-77 A c x) \sqrt{a+c x^2}}{3003 c}-\frac{4 a \sqrt{e x} (39 a B-385 A c x) \left (a+c x^2\right )^{3/2}}{9009 c}-\frac{2 \sqrt{e x} (13 a B-165 A c x) \left (a+c x^2\right )^{5/2}}{2145 c}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{7/2}}{15 c}-\frac{16 a^{13/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{39 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a^{13/4} \left (13 \sqrt{a} B-77 A \sqrt{c}\right ) e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3003 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0975198, size = 117, normalized size = 0.29 \[ \frac{2 \sqrt{e x} \sqrt{a+c x^2} \left (5 a^2 A c x \, _2F_1\left (-\frac{5}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )+a^3 (-B) \, _2F_1\left (-\frac{5}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{a}\right )+B \left (a+c x^2\right )^3 \sqrt{\frac{c x^2}{a}+1}\right )}{15 c \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 381, normalized size = 0.9 \begin{align*} -{\frac{2}{45045\,{c}^{2}x}\sqrt{ex} \left ( -3003\,B{x}^{9}{c}^{5}-3465\,A{x}^{8}{c}^{5}-11739\,B{x}^{7}a{c}^{4}-14245\,A{x}^{6}a{c}^{4}+4620\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{4}c-9240\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{4}c+780\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{4}-16809\,B{x}^{5}{a}^{2}{c}^{3}-22715\,A{x}^{4}{a}^{2}{c}^{3}-9633\,B{x}^{3}{a}^{3}{c}^{2}-11935\,A{x}^{2}{a}^{3}{c}^{2}-1560\,Bx{a}^{4}c \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \sqrt{e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 40.7751, size = 299, normalized size = 0.74 \begin{align*} \frac{A a^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{A a^{\frac{3}{2}} c \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{A \sqrt{a} c^{2} \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{5} \Gamma \left (\frac{15}{4}\right )} + \frac{B a^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{2} \Gamma \left (\frac{9}{4}\right )} + \frac{B a^{\frac{3}{2}} c \left (e x\right )^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{e^{4} \Gamma \left (\frac{13}{4}\right )} + \frac{B \sqrt{a} c^{2} \left (e x\right )^{\frac{13}{2}} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{6} \Gamma \left (\frac{17}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \sqrt{e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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